integral from-5 to 0 of (sqrt(25-x^2))

Lösung

\int_{- 5}^{0} (25 - x^{2}) d x

\frac{25 π}{4}

Dezimale

19.63495 \dots

Schritte zur Lösung

\int_{- 5}^{0} 25 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{2}}^{0} 25 cos^{2} (u) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 25 \cdot \int_{- \frac{π}{2}}^{0} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 25 \cdot \int_{- \frac{π}{2}}^{0} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 25 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{0} 1 + cos (2 u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 25 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{0} 1 d u + \int_{- \frac{π}{2}}^{0} cos (2 u) d u)

\int_{- \frac{π}{2}}^{0} 1 d u = \frac{π}{2}

\int_{- \frac{π}{2}}^{0} cos (2 u) d u = 0

= 25 \cdot \frac{1}{2} (\frac{π}{2} + 0)

Vereinfache

25 \cdot \frac{1}{2} (\frac{π}{2} + 0) : \frac{25 π}{4}

= \frac{25 π}{4}

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